pipe bend analysis

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Offshore Mechanical and Arctic Engineering, July 11-16, 1999 BENDING MOMENT CAPACITY OF PIPES Søren Hauch and Yong Bai American Bureau of Shipping Offshore Technology Department Houston, Texas USA ABSTRACT In most modern pipeline design, the required minimum wall thickness is determined based on a maximum allowable hoop stress under design pressure. This is an efficient way to come up with an initial wall thickness design, based on the assumption that pressure will be the governing load. However, a pipeline may be subjected to additional loads due to installation, seabed contours, impacts and high-pressure/high- temperature operating conditions for which the bending moment capacity is often the limiting parameter. If in-place analyses for the optimal route predict that the maximum allowable moment to a pipeline is going to be exceeded, it will be necessary to either increase the wall thickness or, more conventionally, to perform seabed intervention to reduce the bending of the pipe. In this paper the bending moment capacity for metallic pipes has been investigated with the intention of optimising the cost effectiveness in the seabed intervention design without compromising the safety of the pipe. The focus has been on the derivation of an analytical solution for the ultimate load carrying capacity of pipes subjected to combined pressure, longitudinal force and bending. The derived analytical solution has been thoroughly compared against results obtained by the finite element method. The result of the study is a set of equations for calculating the maximum allowable bending moment including proposed safety factors for different target safety levels. The maximum allowable moment is given as a function of initial out-of- roundness, true longitudinal force and internal/external overpressure. The equations can be used for materials with isotropic as well as an-isotropic stress/strain characteristics in the longitudinal and hoop direction. The analytical approach given herein may also be used for risers and piping if safety factors are calibrated in accordance with appropriate target safety levels. Keywords: Local buckling, Collapse, Capacity, Bending, Pressure, Longitudinal force, Metallic pipelines and risers. NOMENCLATURE A Area OMAE’99, PL-99-5033 Hauch & Bai 1

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Page 1: Pipe Bend Analysis

Offshore Mechanical and Arctic Engineering, July 11-16, 1999

BENDING MOMENT CAPACITY OF PIPES

Søren Hauch and Yong Bai

American Bureau of ShippingOffshore Technology Department

Houston, TexasUSA

ABSTRACT

In most modern pipeline design, the required minimum wall thickness is determined based on a maximum allowable hoop stress under design pressure. This is an efficient way to come up with an initial wall thickness design, based on the assumption that pressure will be the governing load. However, a pipeline may be subjected to additional loads due to installation, seabed contours, impacts and high-pressure/high-temperature operating conditions for which the bending moment capacity is often the limiting parameter. If in-place analyses for the optimal route predict that the maximum allowable moment to a pipeline is going to be exceeded, it will be necessary to either increase the wall thickness or, more conventionally, to perform seabed intervention to reduce the bending of the pipe.

In this paper the bending moment capacity for metallic pipes has been investigated with the intention of optimising the cost effectiveness in the seabed intervention design without compromising the safety of the pipe. The focus has been on the derivation of an analytical solution for the ultimate load carrying capacity of pipes subjected to combined pressure, longitudinal force and bending. The derived analytical solution has been thoroughly compared against results obtained by the finite element method.

The result of the study is a set of equations for calculating the maximum allowable bending moment including proposed safety factors for different target safety levels. The maximum allowable moment is given as a function of initial out-of-roundness, true longitudinal force and internal/external overpressure. The equations can be used for materials with isotropic as well as an-isotropic stress/strain characteristics in the longitudinal and hoop direction. The analytical approach given herein may also be used for risers and piping if safety factors are calibrated in accordance with appropriate target safety levels.

Keywords: Local buckling, Collapse, Capacity, Bending, Pressure, Longitudinal force, Metallic pipelines and risers.

NOMENCLATURE

A AreaD Average diameterE Young’s modulusF True longitudinal forceFl Ultimate true longitudinal forcef0 Initial out-of-roundnessM MomentMC Bending moment capacityMp Ultimate (plastic) momentp Pressurepc Characteristic collapse pressurepe External pressurepel Elastic collapse pressurepi Internal pressurepl Ultimate pressurepp Plastic collapse pressurepy Yield pressurer Average pipe radiusSMTS Specified Minimum Tensile StrengthSMYS Specified Minimum Yield Strengtht Nominal wall thickness Strength anisotropy factor

Distance to cross sectional mass centreC Condition load factorR Strength utilisation factor Curvature Poisson’s ratioh Hoop stresshl Limit hoop stress for pure pressurel Longitudinal stressll Limit longitudinal stress for pure longitudinal force Angle from bending plane to plastic neutral axis

OMAE’99, PL-99-5033 Hauch & Bai 1

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Offshore Mechanical and Arctic Engineering, July 11-16, 1999

INTRODUCTION

Nowadays design of risers and offshore pipelines is often based on a Limit State design approach. In a Limit State design, all foreseeable failure scenarios are considered and the system is designed against the failure mode that is most critical to structural safety. A pipe must sustain installation loads and operational loads. In addition external loads such as those induced by waves, current, uneven seabed, trawl-board impact, pullover, expansion due to temperature changes etc need to be considered. Experience has shown that the main load effect on offshore pipes is bending combined with longitudinal force while subjected to external hydrostatic pressure during installation and internal pressure while in operation. A pipe subjected to increased bending may fail due to local buckling/collapse or fracture, but it is the local buckling/collapse Limit State that commonly dictates the design. The local buckling and collapse strength of metallic pipes has been the main subject for many studies in offshore and civil engineering and this paper should be seen as a supplement to the ongoing debate. See Murphey & Langner (1985), Winter et al (1985), Ellinas (1986), Mohareb et al (1994), Bai et al (1993, 1997) etc.

BENDING MOMENT CAPACITY

The pipe cross sectional bending moment is directly proportional to the pipe curvature, see Figure 1. The example illustrates an initial straight pipe with low D/t (<60) subjected to a load scenario where pressure and longitudinal force are kept constant while an increasing curvature is applied.

Figure 1: Examples of bending moment versus curvature relation.

Different significant points can be identified from the moment-curvature relationship. When applying curvature to a pipe, it will first be subjected to global deformation inside the material’s elastic range and no permanent change in shape is seen. By global deformation is here meant a deformation that can be looked upon as uniform over a range larger than 3-4 times the pipe diameter. After the LINEAR LIMIT of the pipe material has been reached the pipe will no longer return to its initial shape after unloading, but the deformation will still be characterised as global. If the curvature is increased further, material or geometrical imperfections will initiate ONSET OF LOCAL BUCKLING. Imperfections in geometry and/or material may influence where and at which curvature the onset of local buckling occurs, but will for all practical use, as long as they are small, not influence the ULTIMATE MOMENT CAPACITY significantly. After the

onset of local buckling has occurred, the global deformation will continue, but more and more of the applied bending energy will be accumulated in the local buckle which will continue until the ultimate moment capacity is reached. At this point, the maximum bending resistance of the pipe is reached and a geometrical collapse will occur if the curvature is additionally increased. Until the point of START OF CATASTROPHIC CAPACITY REDUCTION has been reached, the geometric collapse will be “slow” and the changes in cross sectional area negligible. After this point, material softening sets in and the pipe cross section will collapse. For pipes that in addition to bending is subjected to longitudinal force and/or pressure close to the ultimate capacity, start of catastrophic capacity reduction occurs immediately after the ultimate moment capacity has been reached. The moment curvature relationship for these load conditions will be closer to that presented by the dashed line in Figure 1.

The moment curvature relationship provides information necessary for design against failure due to bending. Depending on the function of the pipe, any of the points described above can be used as design limit. If the pipe is part of a carrying structure, the elastic limit may be an obvious choice as the design limit. However, for pipelines and risers where the global shape is less important, this criterion will be overly conservative due to the significant resources in the elastic-plastic range. Higher design strength can therefore be obtained by using design criteria based on the stress/strain levels reached at the point of onset for local buckling or at the ultimate moment capacity. For displacement-controlled configurations, it can even be acceptable to allow the deformation of the pipe to continue into the softening region (not in design). The rationale of this is the knowledge of the carrying capacity with high deformations combined with a precise prediction of the deformation pattern and its amplitude.

The moment capacity for metallic pipes is a function of many parameters and the most common are listed below in arbitrary sequence:

Diameter over wall thickness ratio Material stress-strain relationship Material imperfections Welding (Longitudinal as well as circumferential) Initial out-of-roundness Reduction in wall thickness due to e.g. corrosion Cracks (in pipe and/or welding) Local stress concentrations due to e.g. coating Additional loads and their amplitude Temperature

The focus of this study has been the development of an equation to prediction the ultimate moment capacity of pipes. The equation is to account for initial out-of-roundness, longitudinal force and internal/external overpressure for materials with either isotropic or an-isotropic characteristics in longitudinal and hoop direction. Solutions obtained from both analytical expressions and by the finite element method are described in this paper and the results covers a diameter over wall thickness ratio from 10 to 60. The remaining parameters given in the list may also be of some importance in the design of pipelines, but the main parameters will generally be those that are studied in this paper.

OMAE’99, PL-99-5033 Hauch & Bai 2

Ultimate moment capacity

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Offshore Mechanical and Arctic Engineering, July 11-16, 1999

FAILURE MODES

As pointed out in the previous section the ultimate moment capacity is highly dependent on the amount of longitudinal force and pressure loads and for cases with high external pressure also initial out-of-roundness. To clarify the approach used in the development of the analytical equations and to give a better understanding of the obtained results, characteristics of the ultimate strength for pipes subjected to single loads and combined loads are discussed below.

The cross sectional deformations just before failure of pipes subjected to single loads are shown in Figure 2.

Figure 2: Pipe cross sectional deformation of pipes subjected to single loads.

PURE BENDING

A pipe subjected to increasing pure bending will fail as a result of increased ovalisation of the cross section and reduced slope in the stress-strain curve. Up to a certain level of ovalisation, the decrease in moment of inertia will be counterbalanced by increased pipe wall stresses due to strain hardening. When the loss in moment of inertia can no longer be compensated for by the strain hardening, the moment capacity has been reached and catastrophic cross sectional collapse will occur if additional bending is applied. For low D/t, the failure will be initiated on the tensile side of the pipe due to stresses at the outer fibres exceeding the limiting longitudinal stress. For D/t higher than approximately 30-35, the hoop strength of the pipe will be so low compared to the tensile strength that the failure mode will be an inward buckling on the compressive side of the pipe. The geometrical imperfections (excluding corrosion) that are normally allowed in pipeline design will not significantly influence the moment capacity for pure bending, and the capacity can be calculated as, SUPERB (1996):

( )

where D is the average pipe diameter, t the wall thickness and SMYS the Specified Minimum Yield Strength.

represents the average

longitudinal cross sectional stress at failure as a function of the diameter over wall thickness ratio. The average pipe diameter is

conservatively used in here while SUPERB used the outer diameter.

PURE EXTERNAL PRESSURE

Theoretically, a circular pipe without imperfections should continue being circular when subjected to increasing uniform external pressure. However, due to material and/or geometrical imperfections, there will always be a flattening of the pipe, which with increased external pressure will end with a total collapse of the cross section. The change in out-of-roundness, caused by the external pressure, introduces circumferential bending stresses, where the highest stresses occur respectively at the top/bottom and two sides of the flattened cross-section. For low D/t ratios, material softening will occur at these points and the points will behave as a kind of hinge at collapse. The average hoop stress at failure due to external pressure changes with the D/t ratio. For small D/t ratios, the failure is governed by yielding of the cross section, while for larger D/t ratios it is governed by elastic buckling. By elastic buckling is meant that the collapse occurs before the average hoop stress over the cross section has reached the yield stress. At D/t ratios in-between, the failure is a combination of yielding and elastic collapse.

Several formulations have been proposed for estimating the external collapse pressure, but in this paper, only Timoshenko’s and Haagsma’s equations are described. Timoshenko’s equation, which gives the pressure at beginning yield in the extreme fibres, will in general represent a lower bound, while Haagsma’s equation, using a fully plastic yielding condition, will represent an upper bound for the collapse pressure. The collapse pressure of pipes is very dependent on geometrical imperfections and here in special initial out-of-roundness. Both Timoshenko’s and Haagsma’s collapse equation account for initial out-of-roundness inside the range that is normally allowed in pipeline design.

Timoshenko’s equation giving the pressure causing yield at the extreme pipe fibre:

( )

where:

pel = ( )

pp = ( )

and:pc = Characteristic collapse pressuref0 = Initial out-of-roundness, (Dmax-Dmin)/DD = Average diametert = Wall thicknessSMYS = Specified Minimum Yield Strength, hoop directionE = Young’s Module = Poisson’s ratio

It should be noted that the pressure ‘pc’ determined in accordance to Eq. (2) is lower than the actual collapse pressure of the pipe and

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it becomes equal to the latter only in the case of a perfectly round pipe. Hence, by using ‘pc’ calculated from Eq. (2) as the ultimate value of pressure, the results will normally be on the safe side (Timoshenko and Gere, 1961).

Haagsma’s equation giving the pressure at which fully plastic yielding over the wall thickness occurs can be expressed as:

( )

and represent the theoretical upper bound for the collapse pressure. For low D/t, the collapse pressure will be closer to the collapse pressure calculated by Haagsma’s equation than that calculated by Timoshenko’s equation (Haagsma and Schaap, 1981).

The use of Timoshenko’s and Haagsma’s equations relates specifically to pipes with initially linear elastic material properties where the elastic collapse pressure can be derived from classical analysis. This would be appropriate for seamless pipes or for pipes that have been subjected to an annealing process. However, for pipes fabricated using the UO, TRB or UOE method there are significant non-linearity’s in the material properties in the hoop direction, due to residual strains and the Bauschinger effect. These effects may be accounted for by introducing a strength reduction factor to the plastic collapse pressure term given by Eq. (4). In this study no attempt has been given to this reduction factor, but according to DNV 2000 the plastic collapse pressure is to be reduced with 7% for UO and TRB pipes and with 15% for UOE pipes.

PURE INTERNAL PRESSURE

For Pure internal pressure, the failure mode will be bursting of the cross-section. Due to the pressure, the pipe cross section expands and the pipe wall thickness decreases. The decrease in pipe wall thickness is compensated for by an increase in the hoop stress. At a certain pressure, the material strain hardening can no longer compensate for the pipe wall thinning and the maximum internal pressure has been reached. The bursting pressure can in accordance with API (1998) be given as:

( )

where is the hoop stress at failure.

PURE TENSION

For pure tension, the failure of the pipe, as for bursting, will be a result of pipe wall thinning. When the longitudinal tensile force is increased, the pipe cross section will narrow down and the pipe wall thickness decrease. At a certain tensile force, the cross sectional area of the pipe will be reduced so much that the maximum tensile stress for the pipe material is reached. An additional increase in tensile force will now cause the pipe to fail. The ultimate tensile force can be calculated as:

( )

where A is the cross sectional area and

the longitudinal tensile stress at failure.

PURE COMPRESSION

A pipe subjected to increasing compressive force will be subjected to Euler buckling. If the compressive force is further increased, the pipe will finally fail due to local buckling. If the pipe is restrained except for in the longitudinal direction, the maximum compressive force may be taken as:

( )

where A is the cross sectional area and

the longitudinal compressive stress at failure.

COMBINED LOADS

For pipes subjected to single loads, the failure is, as described above, dominated by either longitudinal or hoop stresses. This in-teraction can, neglecting the radial stress component and the shear stress components, be described as:

( )

where l is the applied longitudinal stress, h the applied hoop stress and ll and hl the limit stress in their respective direction. The limit stress may differ depending on whether the applied load is compressive or tensile. is a strength anisotropy factor depending on the ratio between the limit stress in the longitudinal and hoop direction respectively. The following definition for the strength anisotropy factor has been suggested by the authors of this paper for external and internal overpressure respectively:

( )

( )

For pipes under combined pressure and longitudinal force, Eq. (9) may be used to find the pipe strength capacity. Alternatives to Eq. (9) are Von Mises, Tresca’s, Hill’s and Tsai-Hill’s yield condition. Experimental tests have been performed by e.g. Corona and Kyriakides (1988). For combined pressure and longitudinal force, the failure mode will be similar to the ones for single loads.

In general, the ultimate strength interaction between longitudinal force and bending may be expressed by the fully plastic interaction curve for tubular cross-sections. However, if D/t is higher than 35, local buckling may occur at the compressive side, leading to a failure slightly inside the fully plastic interaction curve, Chen and Sohal (1988). When tension is dominating, the pipe capacity will be higher than the fully plastic condition due to tensile and strain-hardening effects.

As indicated in Figure 2, pressure and bending both lead to a cross sectional failure. Bending will always lead to ovalisation and finally collapse, while pipes fails in different modes for external and internal overpressure. When bending is combined with external overpressure, both loads will tend to increase the

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Offshore Mechanical and Arctic Engineering, July 11-16, 1999

ovalisation, which leads to a rapid decrease in capacity. For bending combined with internal overpressure, the two failure modes work against each other and thereby “strengthen” the pipe. For high internal overpressure, the collapse will always be initiated on the tensile side of the pipe due to stresses at the outer fibres exceeding the material limit tensile stress. On the compressive side of the pipe, the high internal pressure will tend to initiate an outward buckle, which will increase the pipe diameter locally and thereby increase the moment of inertia and the bending moment capacity of the pipe. The moment capacity will therefore be expected to be higher for internal overpressure compared with a corresponding external pressure.

ADDITIONAL FAILURE MODE

In addition to the failure modes described above, fracture is a possible failure mode for all the described load conditions. In particular for the combination of tension, high internal pressure and bending, it is important to check against fracture because of the high tensile stress level at the limit bending moment. The fracture criteria are not included in this paper, but shall be addressed in design.

EXPRESSION FOR ULTIMATE MOMENT CAPACITY

In the following section, an analytical solution to the ultimate moment capacity for pipes subjected to combined loads is derived. To keep the complexity of the equations on a reasonable level, the following assumptions have been made:

The pipe is geometrically perfect except for initial out-of-roundness

The cross sectional geometry does not change before the ultimate moment is reached

The cross sectional stress distribution at failure can be idealised in accordance with Figure 3.

The interaction between limit longitudinal and hoop stress can be described in accordance with Eq. (9)

FAILURE LIMIT STRESS

The pipe wall stress condition for the bending moment Limit State can be considered as that of a material under bi-axial loads. It is in here assumed that the interaction between average cross sectional longitudinal and hoop stress at pipe failure can be described by Eq. (12). The failure limit stresses are here, neglecting the radial stress component and the shear stress components, described as a function of the longitudinal stress ‘l’, the hoop stress ‘h’ and the failure limit stresses under uni-axial load ‘ll’ and ‘hl’ in their respective direction. The absolute value of the uni-axial limit stresses, which should not mistakenly be taken as the yield stress, are to be used, while the actual stresses are to be taken as positive when in tension and negative when in compression.

( )

where is a strength anisotropy factor depending on the hl/ll

ratio.

Solving the second-degree equation for the longitudinal stress ‘l’ gives:

( )

comp is now defined as the limit longitudinal compressive stress in the pipe wall and thereby equal to l as determined above with the negative sign before the square root. The limit tensile stress tens is accordingly equal to l with the positive sign in front of the square root.

( )

( )

THE BENDING MOMENT

The bending moment capacity of a pipe can by idealising the cross sectional stress distribution at failure in accordance with Figure 3., be calculated as:

( )

Where Acomp and Atens are respectively the cross sectional area in compression and tension, their mass centres distance to the pipe mass centre and the idealised stress level.

Figure 3: Pipe cross section with stress distribution diagram (dashed line) and idealised stress diagram for plastified cross section (full line).

For a geometrical perfect circular pipe, the area in compression and tension can approximately be calculated as:

( )

( )

The distance from the mass centre to the pipe cross section centre can be taken as:

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Offshore Mechanical and Arctic Engineering, July 11-16, 1999

( )

( )

where r is the average pipe wall radius and the angle from the bending plan to the plastic neutral axis. The plastic neutral axis is defined as the axis at which the longitudinal pipe wall stresses change from tensile to compressive, see Figure 3.

Inserting Eq. (17) to (20) in Eq. (16) gives the bending moment capacity as:

( )

LOCATION OF FULLY PLASTIC NEUTRAL AXIS

The angle to the fully plastic neutral axis from the plane of bending can be deduced from the following simplified expression for the true longitudinal pipe wall force:

( )

where the area in compression Acomp is calculated as:

( )

and the area in tension Atens as;

( )

Giving:

( )

Solving Eq. (25) for gives:

( )

or

( )

FINAL EXPRESSION FOR MOMENT CAPACITY

Substituting the expression for the plastic neutral axis, Eq. (27), into the equation for the moment capacity, Eq. (21) gives:

( )

and substituting the expression for tensile and compressive stress, Eq. (14) and (15) into Eq. (28) gives the final expression for the bending moment capacity:

( )or alternatively and more useful in design situations:

( )where

MC = Ultimate bending moment capacityMp = Plastic momentp = Pressure acting on the pipepl = Ultimate pressure capacityF = True longitudinal force acting on the pipeFl = True longitudinal ultimate force

When the uni-axial limit stress in the circumferential and longit-udinal direction are taken as the material yield stress and set to ½, Eq. (29) and (30) specialises to that presented by among others Winter et al (1985) and Mohareb et al (1994).

APPLICABLE RANGE FOR MOMENT CAPACITY EQUATION

To avoid complex solutions when solving Eq. (30), the expres-sions under the square root must be positive, which gives the the-oretical range for the pressure to:

( )

where the ultimate pressure pl depends on the load condition and on the ratio between the limit force and the limit pressure.

Since the wall thickness design is based on the operating pressure of the pipeline, this range should not give any problems in the design.

Given the physical limitation that the angle to the plastic neutral axis must be between 0 and 180 degrees, the equation is valid for the following range of longitudinal force:

( )

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where the ultimate loads Fl and pl depend on the load condition and on the ratio between the ultimate true longitudinal force Fl

and the ultimate pressure pl.

For the design of pipelines, this range is normally not going to give any problems, but again, the range may be reduced due to the question of fracture.

FINITE ELEMENT MODEL

This section describes how a pipe section is modelled using the finite element method. The finite element method is a method where a physical system, such as an engineering component or structure, is divided into small sub regions/elements. Each element is an essential simple unit in space for which the behaviour can be calculated by a shape function interpolated from the nodal values of the element. This in such a way that inter-element continuity tends to be maintained in the assemblage. Connecting the shape functions for each element now forms an approximating function for the entire physical system. In the finite element formulation, the principles of virtual work together with the established shape functions are used to transform the differential equations of equilibrium into algebraic equations. In a few words, the finite element method can be defined as a Rayleigh-Ritz method in which the approximating field is interpolated in piece wise fashion from the degree of freedom that are nodal values of the field. The modelled pipe section is subject to pressure, longitudinal force and bending with the purpose of provoking structural failure of the pipe. The deformation pattern at failure will introduce both geometrical and material non-linearity. The non-linearity of the buckling/collapse phenomenon makes finite element analyses superior to analytical expressions for estimating the strength capacity. In order to get a reliable finite element prediction of the buckling/collapse deformation behaviour the following factors must be taken into account:

A proper representation of the constitutive law of the pipe material

A proper representation of the boundary conditions A proper application of the load sequence The ability to address large deformations, large

rotations, and finite strains The ability to model/describe all relevant failure modes

The material definition included in the finite element model is of high importance, since the model is subjected to deformations long into the elasto-plastic range. In the post-buckling phase, strain levels between 10% and 20% are usual and the material definition should therefore at least be governing up to this level. In the present analyses, a Ramberg-Osgood stress-strain relationship has been used. For this, two points on the stress-strain curve are required along with the material Young’s modules. The two points can be anywhere along the curve, and for the present model, Spe-cified Minimum Yield Strength (SMYS) associated with a strain of 0.5% and the Specified Minimum Tensile Strength (SMTS) corresponding to approximately 20% strain has been used. The material yield limit has been defined as approximately 80% of SMYS.

The advantage in using SMYS and SMTS instead of a stress-strain curve obtained from a specific test is that the statistical uncertainty in the material stress-strain relation is accounted for. It is thereby ensured that the stress-strain curve used in a finite element analysis in general will be more conservative than that from a specific laboratory test.

To reduce computing time, symmetry of the problem has been used to reduce the finite element model to one-quarter of a pipe section, see Figure 4. The length of the model is two times the pipe diameter, which in general will be sufficient to catch all buckling/collapse failure modes.

The general-purpose shell element used in the present model ac-counts for finite membrane strains and allows for changes in shell thickness, which makes it suitable for large-strain analysis. The ele-ment definition allows for transverse shear deformation and uses thick shell theory when the shell thickness increases and discrete Kirchoff thin shell theory as the thickness decreases.

Figure 4 shows an example of a buckled/collapsed finite element model representing an initial perfect pipe subjected to pure bending.

Figure 4: Model example of buckled/collapsed pipe section.

For a further discussion and verification of the used finite element model, see Bai et al (1993), Mohareb et al (1994), Bruschi et al (1995) and Hauch & Bai (1998).

ANALYTICAL SOLUTION VERSUS FINITE ELEMENT RESULTS

In the following, the above-presented equations are compared with results obtained from finite element analyses. First are the capacity equations for pipes subjected to single loads compared with finite element results for a D/t ratio from 10 to 60. Secondly the moment capacity equations for combined longitudinal force, pressure and bending are compared against finite element results.

STRENGTH CAPACITY OF PIPES SUBJECTED TO SINGLE LOADS

As a verification of the finite element model, the strength capacities for single loads obtained from finite element analyses are compared against the verified analytical expressions described in the previous sections of this paper. The strength capacity has been compared for a large range of diameter over wall thickness to demonstrate the finite element model’s capability to catch the right failure mode independently of the D/t ratio.

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For all analyses presented in this paper, the average pipe diameter is 0.5088m, SMYS = 450 MPa and SMTS = 530 MPa. In Figure 5 the bending moment capacity found from finite element analysis has been compared against the bending moment capacity equation, Eq. (1). In Figure 6 the limit tensile longitudinal force Eq. (7), in Figure 7 the collapse pressure Eq. (2, 5) and in Figure 8 the bursting pressure Eq. (6) are compared against finite element results. The good agreement presented in figure 5-8 between finite element results and analytical solutions generally accepted by the industry, gives good reasons to expect that the finite element model also give reliable predictions for combined loads.

10 20 30 40 50 600

1

2

3

4

5

6

7x 10

6

Diameter Over Wall Thickness

Ult

imat

e M

omen

t Cap

acit

y

X = FE results___ = Analytical

Figure 5: Moment capacity as a function of diameter over wall thickness for a pipe subjected to pure bending.

10 20 30 40 50 600.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

7

Diameter Over Wall Thickness

Ult

imat

e Tr

ue L

ongi

tudi

nal F

orce

X

= FE results___ = Analytical

Figure 6: Limit longitudinal force as a function of diameter over wall thickness for a pipe subjected to pure tensile force.

10 20 30 40 50 600

1

2

3

4

5

6

7

8

9x 10

7

Diameter Over Wall Thickness

Col

laps

e P

ress

ure

X = FE results___ = Haagsma- - - = Timoshenko

Figure 7: Collapse pressure as a function of diameter over wall thickness for a pipe subjected to pure external overpressure. Initial out-of-roundness f0 equal to 1.5%.

10 20 30 40 50 601

2

3

4

5

6

7

8

9

10x 10

7

Diameter Over Wall Thickness

Bur

st P

ress

ure

X = FE results___ = Analytical

Figure 8: Bursting pressure as a function of diameter over wall thickness for a pipe subjected to pure internal overpressure.

STRENGTH CAPACITY FOR COMBINED LOADS

For the results presented in Figures 9-14 the following pipe dimensions have been used:

D/t = 35fo = 1.5 %SMYS = 450 MPaSMTS = 530 MPa = 1/5 for external overpressure and 2/3 for

internal overpressure

Figures 9 and 10 show the moment capacity surface given by Eq. (31). In Figure 9, the moment capacity surface is seen from the external pressure, compressive longitudinal force side and in Figure 10 it is seen from above. Figures 5 to 8 have demonstrated that for single loads, the failure surface agrees well with finite element analyses for a large D/t range. To demonstrate that Eq. (31) also agrees with finite element analyses for combined loads, the failure surface has been cut for different fixed values of longitudinal force and pressure respectively as demonstrated in Figure 10 by the full straight lines. The cuts and respective finite

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element results are shown in Figures 11 to 14. In Figure 11 the moment capacity is plotted as a function of pressure. The limit pressure for external overpressure is given by Haagsma’s collapse equation Eq. (5) and the limit pressure for internal overpressure by the bursting pressure Eq. (6). For the non-pressurised pipe, the moment capacity is given by Eq. (1). In Figure 12, the moment capacity is plotted as a function of longitudinal force. The limit force has been given by Eq. (7) and (8). For a given water depth, the external pressure will be approximately constant, while the axial force may vary along the pipe. Figure 13 shows the moment capacity as a function of longitudinal force for an external overpressure equal to 0.8 times the collapse pressure calculated by Haagsma’s collapse equation Eq. (5). Figure 14 again shows the moment capacity as a function of longitudinal force, but this time for an internal overpressure equal to 0.9 times the plastic buckling pressure given by Eq. (4). Based on the results presented in Figures 11 to 14, it is concluded that the analytically deduced moment capacity and finite element results are in good agreement for the entire range of longitudinal force and pressure. However, the equations tend to be a slightly non-conservative for external pressure very close to the collapse pressure. This is in agreement with the previous discussion about Timoshenko’s and Haagsma’s collapse equations.

Figure 9: Limit bending moment surface as a function of pressure and longitudinal force.

Figure 10: Limit bending moment surface as a function of pressure and longitudinal force including cross sections for which comparison between analytical solution and results from finite element analyses has been performed.

-0.5 0 0.5 1

-1

-0.5

0

0.5

1

Pressure / Plastic Collapse Pressure

Mo

men

t /

Pla

stic

Mo

men

t

X = FE results___= Analytical

Figure 11: Normalised bending moment capacity as a function of pressure. No longitudinal force is applied.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

True Longitudinal Force / Ultimate True Longitudinal Force

Mom

ent /

Pla

stic

Mom

ent

X = FE results___ = Analytical

Figure 12: Normalised bending moment capacity as a function of longitudinal force. Pressure equal to zero.

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

True Longitudinal Force / Ultimate True Longitudinal Force

Mom

ent /

Pla

stic

Mom

ent

X = FE results___ = Analytical

Figure 13: Normalised bending moment capacity as a function of longitudinal force. Pressure equal to 0.8 times Haagsma’s collapse pressure Eq. (5).

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-0.5 0 0.5 1 1.5-1

-0.5

0

0.5

1

True Longitudinal Force / Ultimate True Longitudinal Force

Mom

ent /

Pla

stic

Mom

ent

X = FE results___ = Analytical

Figure 14: Normalised bending moment capacity as a function of longitudinal force. Pressure equal to 0.9 times the plastic buckling pressure Eq. (4).

USAGE/SAFETY FACTORS

The local buckling check can be separated into a check for load controlled situations (bending moment) and one for displacement controlled situations (strain level). When no usage/safety factors are applied in the buckling check calculations, the two checks ought to result in the same bending capacity. In design though, usage/safety factors are introduced to account for modelling and input uncertainties. The reduction in bending capacity introduced by the usage factors will not be the same for load and displacement controlled situations. Due to the pipe moment versus strain relationship, a higher allowable strength can be achieved for a given target safety level by using a strain-based criterion than by a moment criterion. In this paper only the allowable bending moment criterion is given. This criterion can be used for both load and displacement controlled situations, but may as mentioned be overly conservative for displacement controlled situations.

The usage factor approach presented in this paper is based on shrinking the failure surface shown in Figures 9 and 10. Instead of representing the bending moment capacity, the surface is scaled to represent the maximum allowable bending moment associated with a given target safety level. The shape of the failure surface given Eq. (30) is dictated by four parameters; the plastic moment Mp, the limit longitudinal force Fl, the limit pressure Pl and the strength anisotropy factor . To shrink the failure surface usage factors are applied to the plastic moment, longitudinal limit force and the limit pressure respectively. The usage factors are functions of modelling, geometrical and material uncertainties and will therefore vary for the three capacity parameters. In general, the variation will be small and for simplification purposes, the most conservative usage factor may be applied to all capacity loads. The strength anisotropy factor is a function of the longitudinal limit force and the limit pressure, but for simplicity, no usage factor has been applied to this parameter. The modelling uncertainty is highly connected to the use of the equation. In the SUPERB (1996) project, the use of the moment criteria is divided into four unlike scenarios; 1) pipelines resting on uneven seabed, 2) pressure test condition, 3) continuous stiff supported pipe and 4) all other scenarios. To account for the variation in modelling uncertainty, a condition load factor C is applied to the plastic

moment and the limit longitudinal force. The pressure, which is a function of internal pressure and water depth, will not be subjected to the same model uncertainty and the condition load factor will be close to one and is presently ignored. Based on the above discussion, the maximum allowable bending moment may be expressed as:

( )where

MAllowable = Allowable bending momentC = Condition load factorR = Strength usage factors

The usage/safety factor methodology used in Eq. (33) ensures that the safety levels are uniformly maintained for all load combina-tions.

In the following guideline for bending strength calculations, the suggested condition load factor is in accordance with the results presented in the SUPERB (1996) report, later used in DNV (2000). The strength usage factors RM, RF and RP are based on comparison with existing codes and the engineering experience of the authors.

GUIDELINE FOR BENDING STRENGTH CALCULATIONS

LOCAL BUCKLING:For pipelines subjected to combined pressure, longitudinal force and bending, local buckling may occur. The failure mode may be yielding of the cross section or buckling on the compressive side of the pipe. The criteria given in this guideline may be used to calculate the maximum allowable bending moment for a given scenario. It shall be noted that the maximum allowable bending moment given in this guideline does not take fracture into account and that fracture criteria therefore may reduce the bending capacity of the pipe. This particularly applies for high-tension / high internal pressure load conditions.

LOAD VERSUS DISPLACEMENT CONTROLLED SITUATIONS:The local buckling check can be separated into a check for load controlled situations (bending moment) and one for displacement controlled situations (strain level). Due to the relation between applied bending moment and maximum strain in pipes, a higher allowable strength for a given target safety level can be achieved by using a strain-based criterion rather than a bending moment criterion. The bending moment criterion can due to this, conservatively be used for both load and displacement controlled situations. In this guideline only the bending moment criterion is given.

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LOCAL BUCKLING AND ACCUMULATED OUT-OF-ROUNDNESS:Increased out-of-roundness due to installation and cyclic operating loads may aggravate local buckling and is to be considered. It is recommended that out-of-roundness, due to through life loads, be simulated using e.g. finite element analysis.

MAXIMUM ALLOWABLE BENDING MOMENT:The allowable bending moment for local buckling under load controlled situations can be expressed as:

whereMAllowable = Allowable bending momentMp = Plastic momentpl = Limit pressurep = Pressure acting on the pipeFl = Limit longitudinal forceF = Longitudinal force acting on the pipe = Strength anisotropy factorC = Condition load factorR = Strength usage factor

STRENGTH ANISOTROPY FACTOR:

for external overpressure

for internal overpressure

If possible, the strength anisotropy factor should be verified by finite element analyses.

PLASTIC (LIMIT) MOMENT:The limit moment may be given as:

whereSMYS = Specified Minimum Yield Strength in

longitudinal directionD = Average diametert = Wall thickness

LIMIT LONGITUDINAL FORCE FOR COMPRESSION AND TENSION:The limit longitudinal force may be estimated as:

whereA = Cross sectional area, which may be

calculated as Dt.SMYS = Specified Minimum Yield Strength in

longitudinal directionSMTS = Specified Minimum Tensile Strength in

longitudinal direction

LIMIT PRESSURE FOR EXTERNAL OVERPRESSURE CONDITION:The limit external pressure ‘pl’ is to be calculated based on:

where

pel =

pp = 1)

f0 = Initial out-of-roundness 2), (Dmax-Dmin)/DSMYS = Specified Minimum Yield Strength in hoop

directionE = Young’s Module = Poisson’s ratio

Guidance note:1) fab is 0.925 for pipes fabricated by the UO precess, 0.85 for

pipes fabricated by the UOE process and 1 for seamless or annealed pipes.

2) Out-of-roundness caused during the construction phase and due to cyclic loading is to be included, but not flattening due to external water pressure or bending in as-laid position.

LIMIT PRESSURE FOR INTERNAL OVERPRESSURE CONDITION:The limit pressure will be equal to the bursting pressure and may be taken as:

where

SMYS = Specified Minimum Yield Strength in hoopdirection

SMTS = Specified Minimum Tensile Strength in hoopdirection

LOAD AND USAGE FACTORS:Load factor C and usage factor R are listed in Table 1.

Table 1: Load and usage factors. Safety Classes

Safety factors

Low Normal High

C

Uneven seabed 1.07 1.07 1.07Pressure test 0.93 0.93 0.93Stiff supported 0.82 0.82 0.82Otherwise 1.00 1.00 1.00

RP Pressure 0.95 0.93 0.90

RF Longitudinal force 0.90 0.85 0.80

RM Moment 0.80 0.73 0.65

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Guidance notes:- Load Condition Factors may be combined e.g. Load

Condition Factor for pressure test of pipelines resting on uneven seabed, 1.070.93 = 1.00

- Safety class is low for temporary phases. For the operating phase, safety class is normal and high for area classified as zone 1 and zone 2 respectively.

CONCLUSIONS

The moment capacity equations in the existing codes are for some load conditions overly conservative and for others non-conservative. This paper presents a new set of design equations that are accurate and simple. The derived analytical equations have been based on the mechanism of failure modes and have been extensively compared with finite element results. The use of safety factors has been simplified compared with existing codes and the target safety levels are in accordance with DNV (2000), ISO (1998) and API (1998). The applied safety factor methodology ensures that the target safety levels are uniformly maintained for all load combinations. It is the hope of the authors that this paper will help engineers in their aim to design safer and more cost-effective pipes.

It is recommended that the strength anisotropy factor be investigated in more detail.

ACKNOWLEDGEMENT

The authors acknowledge their earlier employer formerly J P Kenny A/S now ABB Pipeline and Riser Section for their support and understanding without which this paper would not have been possible.

REFERENCES

API (1998) “Design, Construction, Operation and Maintenance of Offshore Hydrocarbon Pipelines (Limit State Design)”. Bai, Y., Igland, R. and Moan, T. (1993) “Tube Collapse under Combined Pressure, Tension and Bending”, International Journal of Offshore and Polar Engineering, Vol. 3(2), pp. 121-129.

Bai, Y., Igland, R. and Moan, T. (1997) “Tube Collapse under Combined External Pressure, Tension and Bending”, Journal of Marine Structures, Vol. 10, No. 5, pp. 389-410.

Bruschi, R., Monti, P., Bolzoni, G., Tagliaferri, R. (1995), “Finite Element Method as Numerical Laboratory for Analysing Pipeline Response under Internal Pressure, Axial Load, Bending Moment” OMAE’95.

Chen, W. F., and Sohal, I. S. (1988), “Cylindrical Members In Offshore Structures” Thin-Walled Structure, Vol. 6 1988. Special Issue on Offshore Structures, Elsevier Applied Science.

Galambos, T.V. (1998), “Guide to Stability Design Criteria for Metal Structures” John Wiley & Sons.

Corona, E. and Kyriakides, S. (1988), “On the Collapse of Inelastic Tubes under Combined Bending and Pressure”, Int. J. Solids Structures Vol. 24 No. 5. pp. 505-535. 1998.

DNV (2000) Offshore Standard OS-F101 “Submarine Pipeline Systems” Det Norske Veritas, Veritasveien 1, N-1322 Hövik, Norway, January 2000.

Ellinas, C. P., Raven, P.W.J., Walker, A.C. and Davies, P (1986). “Limit State Philosophy in Pipeline Design”, Journal of Energy Resources Technology, Transactions of ASME, January 1986.

Haagsma, S. C., Schaap D. (1981) “Collapse Resistance of Submarine Lines Studied” Oil & Gas Journal, February 1981.

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Hauch, S. and Bai, Y. (1998), “Use of Finite Element Analysis for Local Buckling Design of Pipelines” OMAE’98

Hill, R. (1950), “The mathematical theory of plasticity” Oxford University Press, New York, ISBN 0 19 856162 8.

ISO/DIS 13623 (1998) “Petroleum and Natural Gas Industries – Pipelines Transportation Systems”.

Kyriakides, S and Yeh, M. K. (1985), “Factors Affecting Pipe Collapse” Engineering Mechanics Research Laboratory, EMRL Report No 85/1, A.G.A Catalogue No. L51479 Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin.

Kyriakides, S and Yeh, M. K. (1988), “Plastic Anisotropy in Drawn Metal Tubes” Journal of Engineering for Industry, August 1988, Vol. 110/303.

Kyriakides, S. and Ju, G. T. (1992), “Bifurcation and Localization Instabilities in Cylindrical Shells Under Bending-I-Experiments” Int. J Solids and Structures, Vol. 29, No 9, pp 1117-1142.

Mohareb, M. E., Elwi, A. E., Kulak, G. L. and Murray D. W. (1994), Deformational Behaviour of Line Pipe” Structural Engineering Report No. 202, University of Alberta

Murphey C.E. and Langner C.G. (1985), “Ultimate Pipe Strength Under Bending, Collapse and Fatigue”, OMAE’85.

SUPERB (1996), “Buckling and Collapse Limit State”, December 1996.

Timoshenko, S. P. and Gere, J. M. (1961), “Theory of Elastic Stability”, 3rd Edition, McGraw-Hill International Book Company.

Winter, P.E., Stark J.W.B. and Witteveen, J. (1985), “Collapse Behaviour of Submarine Pipelines”, Chapter 7 of “Shell Structures Stability and Strength” Published by Elsevier Applied Science Publishers, 1985.

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